GEOMETRY OF DOMAINS WITH THE UNIFORM SQUEEZING
PROPERTY
SAI-KEE YEUNG
Abstract We introduce the notion of domains with (a, b)-uniform squeezing property, study various analytic and geometric properties of such domains and show
that they cover many interesting examples, including Teichmüller spaces and Hermitian symmetric spaces of non-compact type. The properties supported by such
manifolds include pseudoconvexity, hyperconvexity, Kähler-hyperbolicity, vanishing
of cohomology groups and quasi-isometry of various invariant metrics. It also leads
to nice geometric properties for manifolds covered by bounded domains and a simple
criterion to provide positive examples to a problem of Serre about Stein properties
of holomorphic fiber bundles.
§0 Introduction
The purpose of this article is to introduce a class of bounded domains in Cn
which on one hand is sufficiently general to include interesting classes of examples
and on the other hand leads to interesting analytic and geometric properties.
Definition 1. Denote by Br (x) a ball of radius r in Cn . Let 0 < a < b < ∞
be positive constants. A bounded domain M in Cn for some n > 1 is said to have
the uniform squeezing property, or more precisely, (a, b)-uniform squeezing property
if there exist constants a and b, such that for each point x ∈ M, there exists an
embedding ϕx : M → Cn with ϕx (x) = 0 and Ba (ϕx (x)) ⊂ ϕx (M ) ⊂ Bb (ϕx (x)).
We call the corresponding coordinate system a uniform squeezing coordinate system
or, more precisely, (a, b)-uniform coordinate system.
The notation is an immediate generalization of the notation of Bers Embedding
(cf. [Ga]). After the appearance of the present paper in Adv. in Math., Kefeng
Liu and Shing-Tung Yau have kindly informed the author that they have named in
the paper [LSY] a similar domain a ‘holomorphic homogeneous regular manifold’,
not concerning the dependence on (a, b). The small overlap between new results of
the present paper and [LSY] lies on part of the first sentence in Theorem 2a about
quasi-isometry of gK , gC and gB , which is also stated in [LSY].
Our paper interests in the dependence on (a, b). All results in the paper can be
estimated effectively in a and b, and is achieved mainly through estimates involving
gKE .
Even though the definition is very simple and appears to be rather restrictive,
it in fact includes lots of interesting examples.
Key Words: bounded domains, pseudoconvexity, Kähler-Einstein metric, Kähler hyperbolic,
moduli space
1991 Mathematics Subject Classification. Primary 32G15, 53C55, 55N99
The author was partially supported by grants from the National Science Foundation and the
National Security Agency.
1
2
SAI-KEE YEUNG
Proposition 1. Examples of bounded domains with the uniform squeezing property
include the followings,
(a). bounded homogeneous domains,
(b). bounded strongly convex domains,
(c). bounded domains which cover a compact Kähler manifold, and
(d). Teichmüller spaces Tg,n of hyperbolic Riemann surfaces of genus g with n
punctures.
We remark that Hermitian symmetric spaces of non-compact type constitute
an important subclass of both (a) and (c). The former follows from the HarishChandra realization of such symmetric spaces, and the latter follows from existence
of cocompact arithemtic lattices associated to the automorphism groups of the
symmetric spaces (cf. [B]).
Our main objective is to show that domains with uniform squeezing properties
support many interesting geometric and analytic properties. The first observation
is about pseudoconvexity of such domains.
Theorem 1. Let M be a domain with the uniform squeezing property. Then the
following conclusions are valid.
(a). The Bergman metric of M is complete.
(b). M is a pseudoconvex domain.
(c). There exists a complete Kähler-Einstein metric on M .
The second observation is about the behavior of invariant metrics on such domains. On a general bounded domain, there are three well-known intrinsic metrics
which are invariant under a biholomorphism, namely, the Kobayashi metric, the
Carathéodory metric and the Bergman metric. There is a fourth one when the
bounded domain is pseudoconvex, viz., the Kähler-Einstein metric. We denote the
metrics by gK , gC , gB and gKE respectively. For a Kähler metric g on M , we denote by Rg its curvature tensor and ∇g the Riemannian connection. For tangent
vectors X1 , · · · , XN , we denote ∇gX1 · · · ∇gXN by ∇gX1 ,···XN . Furthermore, ∇gi1 ,···iN
denotes the covariant derivatives with respect to the coordinate vectors. We also
normalized the Kähler-Einstein metric so that Ric(gKE ) = −2(n + 1).
Theorem 2. Let M be a bounded domain with (a, b)-uniform squeezing property.
(a). The invariant metrics gK , gC , gB and gKE are quasi-isometric. Furthermore
a
gK 6 gC 6 gK
b
a
2b
gK 6 gB 6 ( )2n+2 gK
b
a
a2
b4n−2 nn−1
gK 6 gKE 6 (
)gK
b2 n
a4n−2
(b). There exist constants cngB and cgnKE such that k∇gi1B,···iN RgB kgB 6 cgNB and
gKE
k∇gi1KE
for any covariant derivatives ∇gi1B,···iN and ∇gi1KE
,···iN RgKE kgKE 6 cN
,···in of
gB and gKE respectively.
(c). Let X1 , · · · , XN be N tangent vectors of unit length with respect to a metric
g1 at x ∈ M. Then k∇gX11 ,···XN gB − ∇gX11 ,···XN gKE kg1 6 c, for some constant c
depending on N, where g1 = gB or gKE .
(d). Both of gB and gKE are geometrically finite in the sense that they are complete
with bounded curvature and the injectivity radius is bounded from below uniformly
GEOMETRY OF DOMAINS WITH THE UNIFORM SQUEEZING PROPERTY
3
on M .
(e). Both of gB and gKE are Kähler-hyperbolic.
(f ). M is hyperconvex.
Here we recall that a Kähler manifold (X, ω) is Kähler-hyperbolic if on its universal covering, ω can be written as dh where h is bounded uniformly when measured
with respect to ω. M is hyperconvex if there exists a plurisubharmonic exhaustion
function bounded from above on M.
The followings are some well-known consequences of Kähler-hyperbolicity in Theorem 1 and Theorem 2.
Corollary 1. Let M be a uniformly squeezed manifold. Let g be gB or gKE .
(a) The reduced L2 -cohomology groups of M with respect to g = gB or gKE satisfies
0
i
dim(H(2)
(M )) = ∞ and dim(H(2)
(M )) = 0 for all i > 0.
(b) The first eigenvalues of the Beltrami Laplacian operators ∆gB and ∆gKE on
smooth functions on M with respect to gB and gKE respectively are both bounded
from below by 0.
(c) The volume with respect to either g = gB or gKE of any relatively compact
complex submanifold with boundary N ⊂ M of complex dimension k satisfies
vol2k (N ) 6 C · vol2k−1 (∂N ) for some constant C > 0.
We define a lattice on M to be a discrete group acting properly discontinuously
as biholomorphisms on M.
Corollary 2. Assume that Γ is a torsion-free lattice on M which admits a uniform
squeezing coordinate system. Then a compact quotient N = M/Γ has to be a
projective algebraic variety of general type. A non-compact quotient N = M/Γ
which has finite volume with respect to the invariant Bergman metric has to be a
quasi-projective variety of log-general type.
Another direct consequence of Theorem 2b and a result of Mok-Yau in [MY] is
the following estimates on the growth of Bergman kernel.
Corollary 3. Let M be a bounded domain with the uniform squeezing property.
Denote by d = d(z, ∂Ω) the Euclidean distance of z ∈ Ω from the boundary ∂Ω of
c
Ω. Then K(z, z) > d2 (− log
d)2 for some constant c > 0.
Let us now focus on the applications of the above results to more specific compact
or non-compact manifolds.
Theorem 3. Assume that N is a compact complex manifold of complex dimension
n whose universal covering is a bounded domain in Cn . Then the following properties
hold.
(a). N is projective algebraic.
(b). There exists a Kähler-Einstein metric on N .
(c). (−1)n χ(N ) > 0.
(d). H 0 (N, 2K) is non-trivial, where K is the canonical line bundle on N .
(e). The universal covering of N is Stein.
The result can be considered as a support for a conjecture of Shafarevich, which
states that the universal covering of a projective algebraic variety is holomorphically convex (cf. [Ko]). The assumption is stronger but the projective algebraicity
is obtained for free. On the other hand, it also shows that if we try to formulate
4
SAI-KEE YEUNG
a conjecture for the uniformization of a compact complex manifold by a bounded
domain, it should include topological and analytic conditions such as those stated
in (c) and (d). Properties in (c) is along the line of conjectures of Hopf, Chern and
Singer in Riemannian geometry, a consequence of those is that the Euler characteristic of a compact Riemannian manifold of even dimension 2n with non-positive
Riemannian sectional curvature satisfies (c) (cf. [Gr]). Note that a compact torus
is flat and its Euler characteristic is equal to zero.
As a consequence of a result of Stehlé, Theorem 2f provides the following simple
criterion for positive solutions to a problem of Serre [Se], who asked whether a
holomorphic fiber bundle with Stein base and Stein fibers are Stein.
Corollary 4. Suppose π : T → B is a locally trivial holomorphic fiber space for
which the base B is a Stein space and the fibers satisfy the uniform squeezing properties. Then T is also Stein.
As an application of Theorem 1 and 2 to non-compact complex manifolds of
finite volume with respect to some invariant metric, we consider moduli space of
possibly punctured curves as an example.
Theorem 4. Let g, n > 0 and 2g − 2 + n > 0, so that the complement of n punctures of a compact Riemann surface of genus g gives a hyperbolic Riemann surface.
Let Mg,n be the moduli space of such hyperbolic Riemann surfaces. Let Tg,n be the
corresponding Teichmüller space.
(a). gK , gC , gB and gKE are quasi-isometric on Mg,n
(b). For gB and gKE , any order of covariant derivative of the curvature tensor of
the metric is uniformly bounded on Mg,n . As a consequence, for any set of unit
vectors {X1 , · · · XN } measured with respect to g1 , the difference k∇gX11 ,···XN Rg1 −
∇gX21 ,···XN Rg2 kg1 is bounded for any g1 , g2 chosen among gB , gKE , where ∇gX1 ,···XN
denotes the covariant derivatives of a metric g with respect to the vectors X1 , · · · , Xn .
(c). The Teichmüller space Tg,n is Kähler-hyperbolic with respect to both gB and
gKE .
(d). Tg,n is hyper-convex.
(f ). Mg,n is quasi-projective of log-general type and the Euler-Poincaré characteristic satisfies (−1)n χ(Mg,n ) > 0.
Except for the statements related to the Kähler-hyperbolicity of the KählerEinstein metric and estimates on higher order quasi-isometry of the metrics gB and
gKE , most of the results in Theorem 4 can be obtained for example by combining
results in [Y3] and [Y4], but the proofs there rely on many well-known and diverse
results. In this paper, all these properties except quasi-projectivity of moduli space
of curves are derived solely from the existence of uniform squeezing coordinates,
which is provided classically by the Bers Embedding (cf. [Ga]).
Overall we remark that parts of the results in this paper have been obtained
for some specific examples mentioned in Proposition 1. In particular, Kählerhyperbolicity of locally Hermitian symmetric spaces with respect to the Bergman
metric is explained in [Gr], of bounded homogeneous space with respect to the
Bergman metric is proved in [Do], of moduli space of curves with respect to a
metric constructed by McMullen is proved by McMullen in [Mc], where the metric is also shown to have many nice properties such as geometric finiteness and
GEOMETRY OF DOMAINS WITH THE UNIFORM SQUEEZING PROPERTY
5
quasi-isometry to the Kobayashi metric. The vanishing of the cohomology groups
hi(2) , i < n, for Kähler-hyperbolic manifolds was proved by Gromov in [Gr].
There is a vast amount of literature related to Theorem 4. A precise formula
for the Euler characteristic of the moduli space of curves was given by HarerZagier. Mg,n was first shown to be pseudoconvex or a domain of holomorphy
through the work of Bers and Ehrenpreis. Hyperconvexity of Mg,n had been proved
by Krushkal, and more recently in [Y2] by a different and more geometric way
using results of Wolpert. It follows from classical results of Baily, Deligne-Mumford
and Knudson-Mumford that a moduli space of curves is quasi-projective. Quasiisometry among invariant metrics on Mg,n were obtained through the contributions
of Chen, Liu-Sun-Yau and Yeung. We refer the readers to [W] and [Y4] for more
details.
In a sequel to the present paper, we will explain how the set-up of the paper can
be used to prove subelliptic estimates for solutions to the ∂ equations on uniform
squeezing domains, which include Teichmüller spaces, where the problem remained
open in the past due to a lack of description of the boundary of Bers Embedding.
Parts of the work were finished while the author was visiting the Korea Institute
for Advanced Study, the Osaka University, and the University of Hong Kong. The
author would like to thank Jun-Muk Hwang, Toshiki Mabuchi and Ngaiming Mok
for their hospitality. The author would also like to express his gratitude to the
referee for very helpful comments on the paper.
§1 Terminology and preliminaries
Recall the following standard notations about various convexity of a domain.
A domain in Cn is pseudoconvex if there exists a plurisubharmonic exhaustion.
2
A bounded domain Ω = {z ∈ Cn |r(z) < 0} for some
√ C function r(z) in z =
(z1 , · · · , zn ) is strongly pseudoconvex if the Levi form −1∂∂r > 0 in a neighborhood of ∂Ω. A domain Ω in Cn is hyperconvex if there exists a bounded plurisubharmonic exhaustion function.
A Kähler metric ω on a complex manifold M is said to be Kähler-hyperbolic if
on the universal covering M̃ of M, the pull back of ω can be expressed as dη for
some 1-form η which is bounded uniformly on M̃ with respect to ω.
We say that two metrics g1 and g2 are quasi-isometric, denoted by g1 ∼ g2 , if
there exists a positive constant c such that 1c g1 (v, v) 6 g2 (v, v) 6 cg1 (v, v) for all
holomorphic tangent vectors v.
Let us now recall the various notions of invariant metrics on a general complex
manifold.
For a unit tangent vector v ∈ Tx M on a complex manifold M , the Kobayashi
and Carathéodory semi-metrics are defined respectively as complex Finsler metrics
by
p
gK (x, v)
=
p
gC (x, v)
=
1
inf{ |∃f : Br1 → M holomorphic, f (0) = x, f 0 (0) = v}.
r
1
sup{ |∃h : M → Br1 holomorphic, h(x) = 0, |dh(v)| = 1},
r
6
SAI-KEE YEUNG
where we use Brn = Brn (0) to denote a ball of radius r centered at 0 in Cn . Since we
are considering only bounded domains in Cn , both gK and gC are non-degenerate
complex Finsler metrics.
Consider now Kähler-Einstein metric of constant negative scalar curvature. We
normalize the curvature so that gKE satisfies Ric(gKE ) = −2(n + 1), where ωKE is
the Kähler form associated to gKE . The normalization is chosen so that it agrees
with the one for the hyperbolic metric on BCn of constant holomorphic sectional
curvature −4.
The Bergman pseudometric gB on a general complex manifold M of complex
dimension N is a Kähler pseudometric with local potential given by the coefficients
of the Bergman kernel K(x, x). It is clearly non-degenerate for Tg . gB can be
interpreted in the following way.
Let f be a L2 -holomorphic N -form on M , where dimC M = N. In terms of local
coordinates (z1 , · · · , zN ) on a coordinate chart U , let eKM = dz 1 ∧· · ·∧dz N be a local
basis of the canonical line bundle KM on U . We can write f as fU eKM on U . Let
0
fi , i ∈ N be an orthonormal basis of L2 -sections in H(2)
(M, KM ). Note that from
conformality, the choice
is
independent
of
the
metric
on
M . The Bergman kernel
P
P
is given by K(x, x) = i fi ∧ fi . Let KU (x, x) = i fU,i fU,i be the coefficient of
K(x, x) in terms of the local coordinates. The Bergman metric is given by a Kähler
form
X
√
√
1
ωB = −1∂∂ log KU (x, x) = −1
(fi ∂fj − fj ∂fi ) ∧ (fi ∂fj − fj ∂fi ),
KU (x, x)2 i<j
which is clearly independent of the choice of a basis and U . As the Bergman kernel
is independent of basis, for each fixed point x ∈ M,
KU (x, x) =
sup
|fU (x)|2 ,
0 (M,K ),kf k=1
f ∈H(2)
M
where k · k stands for the L2 -norm. We may assume that supf ∈H 0
(M,KM ) |fU (x)|
|fx,U (x)|2 . Using
(2)
0
H(2)
(M, K)
is realized by fx ∈
with kfx k = 1 so that KU (x, x) =
the fact that the Bergman kernel is independent of the choice of a basis again and
letting V ∈ Tx M,
1
ωB (V, V̄ ) =
sup
|V (fU )|2 .
|fx,U (x)|2 f ∈H 0 (M,KM ),kf k=1,f (x)=0
(2)
∂
∂z i .
Consider in particular V =
We may also assume that the supremum for
0
| ∂z∂ i (fU )|2 among all f ∈ H(2)
(M, KM ), kf k = 1, f (x) = 0 is achieved by gi,x ∈
0
H(2)
(M, KM ) of L2 -norm 1. Hence supf ∈H 0 (M,KM ),kf k=1 | ∂z∂ i fU |2 = | ∂z∂ i gi,x,U (x)|2 .
(2)
To simplify our notation, we may simply write
ωB (
∂
∂
1
,
)=
∂z i ∂z i
|fx (x)|2
sup
0 (M,K ),kf k=1,f (0)=0
f ∈H(2)
M
|
| ∂ i gi,x (x)|2
∂
(f )|2 = ∂z
,
i
∂z
|fx (x)|2
since the expression is clearly independent of the choice of U and metric on eU .
Finally let us include here two regularity estimates required for later calculations
for the convenience of the readers. We denote by Wk,p and Ck,α the spaces of
functions on Ba which are bounded with respect to the Sobolev norm k · kk,p and
Hölder norm | ·|k,α on Ba (x) respectively. We refer the readers to [GT] for standard
notations.
GEOMETRY OF DOMAINS WITH THE UNIFORM SQUEEZING PROPERTY
7
Proposition 2. (cf.[GT], page 235, 90) Let Ω0 ⊂⊂ Ω be bounded domains in Rn
with C ∞ boundary. Let L be a second order linear differential operator defined by
Lu = aij (x)Dij u + bi (x)Di u + c(x)u
with sums over repeated indices. Let u be a strong solution of the equation Lu = f.
(a) (Calderon-Zygmund estimates) Suppose that u ∈ H1 (Ω) is a strong solution of
Lu = f with f ∈ L2 (Ω). Assume that for aij , bi , c ∈ C 0 (Ω̄),
aij vi vj
≥
|aij |, |bi |, |c| 6
λ|v|2
∀v ∈ Rn ,
Λ.
Then
kuk2,2,Ω0 6 C1 (kuk0,2,Ω + kf k0,2,Ω )
with constant C1 depending on n, p, λ, Λ, Ω0 , Ω and the moduli of continuity of aij
on Ω0 .
(b) (Schauder estimates) Suppose f ∈ C α (Ω̄) and
|aij |α,Ω , |bi |α,Ω , |c|α,Ω 6 Λ.
Then
|u|2,α,Ω0 6 C2 (|u|0,α + |f |0,α,Ω ),
with constant C2 depending on n, α, λ, Λ, Ω and Ω0 .
In our application, we will always assume that Ω = Ba (0) and Ω0 = Ba/2 (0) for
a fixed value a, after identifying an arbitrary point on the manifold to the origin
with respect to a uniform squeezing coordinate system. We are interested in the
estimates of the bounds and use it to show uniform bound over the manifold of our
interest instead of regularity, which is already known for general elliptic equations.
§2 Pseudoconvexity and related properties
Throughout this section and §3, §4, we let M be a bounded domain with uniform
squeezing coordinates.
Lemma 1. The Bergman metric gB on M is a well-defined complete Kähler metric.
Furthermore, gB is quasi-isometric to gK as a complex Finsler metric.
Proof The (1, 1)-form ωB defined in §1 is only semi-definite in general. We need
to show that it is in fact positive definite and gives rise to a complete metric in our
situation.
The Kähler form ω of the Bergman metric is given by
ωB (
| ∂z∂ i gi,x (x)|2
∂
∂
,
)
=
,
∂z i ∂z i
|fx (x)|2
where fx is a function with L2 -norm ktk = 1 realizing the supremum of |f (x)| among
0
L2 -holomorphic functions f ∈ H(2)
(M ), kf k = 1 on T , and gi,x is a holomorphic
∂
0
(M ), kf k = 1, f (x) = 0.
function realizing supremum of | ∂zi (f )|2 among all f ∈ H(2)
8
SAI-KEE YEUNG
From assumption Ban (x) ⊂ M ⊂ Bbn (x), where Brn (x) denotes a complex ball of
radius r centered at x identified with 0 in Cn . Let volo denote the Euclidean volume
on Cn . Clearly from the Mean Value Inequality
R
R
2
n (x) |fx |
|fx |2
1
Ba
2
M
(fx (x)) 6
6
= 2n
.
n
volo (Br (x))
volo (Ban (x))
a volo (B1n )
The constant function h1 (x) = 1 satisfies h1 (1) = 1 and
kh1 k2 = volo (M ) 6 volo (Bbn ) = b2n volo (B1n ).
Hence |fx (x)| ≥
1
1
[(b)2n volo (B1n )] 2
[
. We conclude that
1
1
1
1
] 2 ≥ |fx (x)| ≥ [ 2n
]2 .
a2n volo (B1n )
b volo (B1n )
Let Vi be the complex line generated by ∂z∂ i in Cn . Then from Generalized
Cauchy Inequality and Mean Value Inequality,
R
R
1
1
|gi,x (y)|dy
[ ∂(B na (x))∩Vi |gi,x (y)|2 dy] 2 [2π a2 ] 2
∂(B n
a (x))∩Vi
∂
2
2
| i gi,x (x)| 6
6
∂z
2π( a2 )2
2π( a2 )2
R
R
1
1
[ ∂(B na (x))∩Vi dy B na (y) |gi,x (w)|2 dvolo (w)] 2 [πa] 2
2
2
6
1
2π( a2 )2 [volo (B na )] 2
2
R
R
1
1
[ ∂(B na (x))∩Vi dy M |gi,x (w)|2 dvolo (w)] 2 [πa] 2
1
2
6
6 a
1
1 .
2π( a2 )2 [volo (B na )] 2
( 2 )1+n [volo (B1n )] 2
2
On
function hi,x = zi satisfies ∂z∂ i hi,x = 1 and hi,x (0) = 0.
R the 2other 1hand the
As B n |zi | = n+1 vol(B1n ), we know that
1
Z
Z
1 2n+2
2
2
2n+2
khi,x k 6
b
volo (B1n ).
|zi | = b
|zi |2 6
n
n
n
+
1
Bb
B1
Hence the function ki,x :=
hi,x
khi,x k
satisfies | ∂z∂ i ki,x | =
2
and kki,x k = 1.
We conclude as before that
1
1
( a2 )n+1 [volo (B1n )] 2
≥|
√
n+1
1
[b2n+2 volo (B1n )] 2
, ki,x (0) = 0
√
∂
n+1
g
(x)|
≥
i,x
1 .
n+1
∂z i
b
[volo (B1n )] 2
Combining the above estimates for fx (x) and gi,x (x), we arrive at
r
√
2 2b n
∂
a 1
( ) ≥ gB (x, i ) ≥ n + 1( )n .
a a
∂z
b b
q
Since a 6 gK (x, ∂z∂ i ) 6 b from Ahlfors Schwarz Lemma, we conclude that
r
r
r
√
2b n+1
∂
∂
a n+1
∂
( )
gK (x, i ) ≥ gB (x, i ) ≥ n + 1( )
gK (x, i ).
a
∂z
∂z
b
∂z
As a2 6 gK (x, V ), gK is non-degenerate on M. The earlier argument estimating
gB by gK from below then implies that gB is non-degenerate. Hence gB is a Kähler
metric.
GEOMETRY OF DOMAINS WITH THE UNIFORM SQUEEZING PROPERTY
9
We prove now that gB is complete. If gB is incomplete, it follows that there is
a geodesic γ of finite length l from a fixed point xo ∈ M approaching to a point
y on ∂M. In particular, given any preassigned number > 0, we can choose a
point z on γ so that the distance dB (z, y) = limw→y dB (z, w) 6 . On the other
hand, the above discussions relating gB to gK actually shows that the distance
dB (z, ∂Ba (x)) with respect to the Bergman metric is at least a · k1 . This clearly
leads to a contradiction by choosing < a · k1 . Hence gB is complete.
Lemma 2. M is a pseudoconvex domain.
Proof Fix a realization of M √as a bounded domain Ω in Cn . From the previous
lemma, the Bergman metric −1∂∂ log KB,Ω is positive definite, here KB,Ω =
P
2
i |fi (z)| is the potential of the Bergman metric on Ω expressed in terms of a
unitary basis {fi } of the space of L2 -holomorphic functions on Ω. Clearly, KB,Ω is
a strictly plurisubharmonic function on M. We need only to prove that KB,Ω blows
up along any sequence of points approaching the boundary of Ω.
For a point x ∈ Ω ∼
= M, let us still use the notation ϕx for the uniformizing
coordinate charts for x as defined in the Introduction. Let KB,ϕx (Ω) be the potential
of the Bergman metric on ϕx (Ω). Clearly in terms of the Jacobian of the transition
functions,
KB,Ω = KB,ϕx (Ω) |J(ϕx )|2 .
From the proof of Lemma 1, we know that
KB,ϕx (Ω) (y, y)
=
sup
|f (y)|2
0 (ϕ (Ω)),kf k
f ∈H(2)
x
ϕx (Ω)) =1
≥
1
R
Bb (0)
1
is bounded from below. Hence it suffices for us to prove that |J(ϕx )| blows up for
x approaching ∂Ω.
−1
Recall from definition that ϕ−1
x : ϕx (Ω) → Ω is a biholomorphism and ϕx (0) =
x. We claim that as x → ∂Ω, the smallest eigenvalue µx of the Jacobian matrix
J(ϕ−1
x )|0 at 0 approaches to 0.
To prove the claim, we assume for the sake of proof by contradiction that there
exists a sequence of points xi ∈ Ω with Euclidean distance d(xi , ∂Ω) = i →
0 but µxi > c1 for some constant c1 > 0. First of all, we observe by applying
−1
the generalized Cauchy estimates to ϕ−1
x on Ba (0) that every derivative of ϕx is
bounded from above by some constant independent of x. In particular, all second
derivatives of ϕ−1
with respect to the coordinate vectors on Ba (0) are bounded
x
from above by a constant c2 > 0. Let now `i be a line segment in Ω realizing the
Euclidean distance between xi and ∂Ω, so that for yi ∈ `i ∩ ∂Ω, d(xi , yi ) = i . The
complexification of `i is a complex line `i,C intersecting Ω. After a linear change of
coordinate, we may assume that `i,C is defined by ζ2 = · · · = ζn = 0. We may also
assume that ζ1 = 0 at xi . Writing ζ = t + iu in terms of real and imaginary part,
we may assume without loss of generality that `i lies on the real axis defined by
u = 0 and parametrized by t for 0 6 t 6 i . Hence the end point on ∂Ω is given by
ζi = 0, i > 2, and ζ1 (yi ) = i .
As ϕxi is a biholomorphism, the image `ei := ϕxi (`i ∩ Ω) is a real curve on ϕxi (Ω)
with 0 as an endpoint. Assume that `ei is parametrized by a unit speed parameter
10
SAI-KEE YEUNG
s so that `ei (0) = 0 on ϕxi (Ω). Since ϕ−1
xi (Ba (0)) intersects `i on Ω, we know that
c1
e
the length of `i in ϕxi is greater than a. Let r = min(a, 4c
). Denote by λ(s) the
2
−1
e
minimal eigenvalue of J(ϕx ) at `(s).
λ(0) = µxi from definition. From the Mean
Value Theorem in calculus, it is clear that for 0 6 s 6 r, the minimal eigenvalue at
s satisfies
c1
λ(s) > λ(0) − c2 s > c1 − c2 r > .
2
dt
It follows that ds
> c21 for 0 6 s 6 r. Hence the length of `i is at least c21 r, a
constant independent of xi . Clearly this contradicts the assumption that the length
of `i , which is d(xi , ∂Ω), is i and i → 0 as i → ∞. The claim is proved.
As mentioned above, each eigenvalue of the Jacobian of ϕ−1
x is bounded from
above by a constant c3 for all points x ∈ Ω. Moreover, the smallest of them approaches to 0 as x → ∂Ω. Since the determinant |J(ϕ−1
x )| is just the product of all
−1
the eigenvalues of the J(ϕ−1
),
we
conclude
that
|J(ϕ
x
x | → 0 as x → ∂Ω. Hence
|J(ϕx )| tends to ∞ as x → ∂Ω.
Remark It was pointed out by the referee that the argument essentially showed
that the trace of the Bergman kernel K(x, x) of a uniform squeezing domain was
bounded from below by c/d, where d = d(x, ∂Ω) is the Euclidean distance to the
boundary of Ω and c is a constant. Later on we will see that as a consequence of
Theorem 1 and 2, the estimates can be improved to c/(d2 (− log d)2 ) as stated in
Corollary 3.
We may now complete the proof of Theorem 1.
Proof of Theorem 1 (a) and (b) follow from Lemma 1 and Lemma 2. (c) follows
from the work of Cheng-Yau and Mok-Yau on Kähler-Einstein metrics (cf. [MY]).
§3 Metric properties
We say that we metrics g1 and g2 are equivalent or quasi-isometric on a domain
∆, denoted by g1 ∼ g2 , if there exists a constant c > 0 such that 1c g2 6 g1 6 cg2 .
Proposition 3. The invariant metrics on a uniformly squeezing domain satisfy
gC ∼ gK ∼ gB ∼ gKE . More precisely,
a
gK 6 gC 6 gK ,
b
a
2b
gK 6 gB 6 ( )n+1 gK ,
b
a
a2
b4n−2 nn−1
gK 6 gKE 6
gK .
b2 n
a4n−2
Proof Since the proof is very similar to the proof of Theorem in [Y3], we would
just give a brief outline.
It follows from Ahlfors Schwarz Lemma that gC 6 gK . On the other hand, from
n
n
definition of gK and gC and the inclusions
p Ba (x) ⊂ 1ϕ(M ) ⊂
p Bb (x), we conclude
for any tangent vector v ∈ Tv M that gK (x, v) 6 a and gC (x, v) ≥ b. Hence
gB 6 ab gC . Hence gB > ab from the above discussions.
The upper bound of gB by gK is already given in the proof of Lemma 1. On the
other hand, as observed by Look and Hahn (cf. [H]), it follows by expressing gB in
terms of extremal functions that gB ≥ gC .
GEOMETRY OF DOMAINS WITH THE UNIFORM SQUEEZING PROPERTY
11
To compare gKE and gK , we normalized the Poincaré metric on Br (0) so that
the potential is log(r2 − |z|2 ). The resulting metric is a Kähler-Einstein metric of
Ricci curvature −2n + 2 with constant holomorphic sectional curvature −4 < 0.
Br
Br
Br
Then gKE
(v, v) = r12 = gK
(x, v). It follows from definition that on Br , gK
(0, v) =
Br
Br
1 2
gC (0, v) = [ r ] = gKE (v, v). Ban ⊂ MS ⊂ Bbn . Let us denote the volume form
of g by µ(g). Applying Schwarz Lemma of Mok-Yau [MY] to the first inclusion
n
Ba
M
with respect to the Kähler-Einstein metrics gKE
and gKE
on Ban and M of Ricci
2n+2
curvature − (a)2 and −(2n + 2) respectively, we get
Bn
Bn
M
a
µ(gKE
) 6 µ(gKE
) = µ(gKa )
b
b
Bn
M
)
= ( )2n µ(gKb ) 6 ( )2n µ(gK
a
a
M
Applying Schwarz Lemma of [R] to gKE
which has constant Ricci curvature −(2n+
n
Bb
2) and gKE which has constant holomorphic sectional curvature −4, we conclude
that
1 Bn
a2 B n
a2 M
1 Bbn
M
.
= gKb ≥ 2 gKa ≥ 2 gK
gKE
≥ gKE
n
n
nb
nb
M
M
Let νi > 0, i = 1, . . . , n be the eigenvalues of gKE
with respect to gK
. We conclude
2
a
from the second estimate that νi ≥ nb2 for all i, and from the first statement that
Qn
2
b 2n
b4n−2 nn−1
≥ νi ≥ ba2 n . Hence
i=1 νi 6 ( a ) . It follows that
a4n−2
a2
b
( )4n−2 nn−1 gK ≥ gKE ≥ 2 gK .
a
b n
This concludes the proof of the proposition.
Proposition 4. (a). There exists a constant cgNKE depending only on the order of
gKE
differentiation N such that k∇gi1KE
for any covariant derivatives
,···iN RgKE k 6 cN
gKE
∇i1 ,···iN . Consequently, the curvature tensor of gKE and any order of covariant
derivatives of the curvature tensor is bounded by a uniform constant. Furthermore,
the injectivity radius of gKE is bounded uniformly from below on M .
(b). The same conclusion is true for the Bergman metric gB .
Proof (a). We denote gKE simply by g in this part of proof. For each fixed
point x ∈ M, there exists a uniformizing squeezing coordinate system given by
Ba (0) ⊂ ϕ(M ) ⊂ Bb (0) ⊂ Cn , where ϕ(x) = 0. We would derive our estimates
on such coordinate neighborhoods. By a unitary change of coordinates, we may
assume that gij (x) is diagonal at x = 0. Furthermore, from Lemma 3, we know that
on B a2 (0), gKE ∼ gK ∼ go . Hence in measuring the magnitude of a derivative with
respect to g = gKE , it is up to some uniform constant the same as measuring with
respect to the Euclidean metric go . We need the following technical estimates.
Lemma 3. Let g = gKE be the Kähler-Einstein metric on a domain M with
uniform squeezing properties. Then all the covariant derivatives of the coordinate
vector fields in terms of the uniform squeezing coordinate systems are uniformly
bounded on M by a constant depending on the order of differentiations.
Proof In terms of the uniform squeezing coordinate system, Lemma 3 is equivalent
to the boundedness of any order of derivatives of the metric coefficient gij with
respect to the coordinate vectors.
12
SAI-KEE YEUNG
In the following, we denote by ci , ck,m and c0k,m constants which are independent
of x ∈ M. The Kähler metric satisfies Einstein equation
(0.1)
∂i ∂ j log | det(g)| = cgij
on Ba (x) for x ∈ M.
Note that gij coming from solution of Monge-Ampère equation is smooth from
the standard results in Kähler-Einstein equations (cf. [Au], chapter 7). In fact, it
would also follow from Proposition 2 together with Theorem 3.56 of [Au]. Taking
trace with respect to the Euclidean metric, we get
(0.2)
∆o log | det(g)| = cgoij gij
on Ba (x).
Recall also from Proposition 3 that on Ba/2 (x), gij is quasi-isometric to the
Euclidean metric |(go )ij | = δij , where δij are Kronecker’s delta.
In terms of the usual notion used in [GT], let us denote by Hk = Wk,2 and Ck,α
the spaces of functions on Ba which are bounded with respect to the Sobolev norm
and Hölder norm on Ba (x) respectively. Applying Calderon-Zymund’s estimates in
Proposition 2 to the Einstein equation, we get
k log | det(g)|kH2 6 c1 [k log | det(g)|kH0 + kgoij gij kH0 ] 6 c2 ,
here again we used Proposition 3.
Observe that on each point y ∈ Ba (x), there exists a unitary matrix Ay such that
t
Ay gAy is a diagonal matrix. As | det Ay | = 1 from definition, we may assume that g
Qn
is diagonal at y for our computation involving det(g). Hence | det(g)| = i=1 gii at
y. The Einstein equation gives rise to ∂i ∂ i log | det(g)| = cgii . Let X be any vector
field of unit length coming from linear combination of coordinate vector fields. Let
DX denote derivative in the direction of X. Then by applying DX to the above
equation
∂i ∂ i DX log | det(g)| = cDX gii = cgii DX log |(gii )|.
Taking the trace by g and summing over all i = 1, . . . , n, we get
n
X
i=1
g ii (y)∂i ∂ i DX log | det(g)| = c
n
X
DX log |(gii )(y)| = DX log | det(g)|(y).
i=1
We obtain ∆g DX log | det(g)| = DX log | det(g)|(y). As gij is uniformly quasi-isometric
to (go )ij , the same Schauder estimate allow us to conclude that
kDX log | det(g)|kH2 6 c3 [kDX log | det(g)|kH0 ] 6 c4
after applying the earlier bound on log | det(g)|kH2 . As X is arbitrary, this implies
k log | det(g)|kH3 6 c4 .
Clearly the bootstrapping argument implies that for each positive integer m, there
exists a constant cm independent of x such that
k log | det(g)|kHm 6 cm .
Applying the Sobolev Estimates to Ba/2 (x), we conclude that
k log | det(g)|kCk,m 6 ck,m
GEOMETRY OF DOMAINS WITH THE UNIFORM SQUEEZING PROPERTY
13
for some constant ck,m independent of x. Applying to equation (0.2), we conclude
that
kgij kCk,m 6 c0k,m
for some constant c0k,m independent of x. This concludes the proof of the Lemma.
We may now complete the proof of (a). We compute in terms of the coordinate
vectors in the uniform squeezing coordinate system. A covariant derivative of the
curvature tensor is a Euclidean derivative modified by an addition term coming
from Christopher symbol, which are linear combinations of the derivatives of the
metric tensor with respect to coordinate vectors. It follows easily from Lemma 3
and by induction that the norm of an n-th order derivative of the metric tensor
with respect to the coordinate vectors is bounded by a constant depending on n
but independent of x. Hence the first statement of (a) follows.
Since the curvature tensor of gKE appears as sum of some second order and first
order derivatives of the metric tensor with respect to coordinate vectors, it is clear
that the curvature tensor is bounded uniformly. Similarly, any order N covariant
derivatives of the curvature are linear combination of expressions involving up to
N +2 order of derivatives of the metric tensor with respect to the coordinate vectors.
We conclude that any such derivative is bounded by a constant depending on N .
From the boundedness in curvature, we conclude immediately that the conjugate
radius is bounded from below by an absolute constant. Furthermore we note that
there exists a > 0 such that a geodesic loop l of length less than based at x
does not exist. Suppose on the contrary such a geodesic exists and the incoming
and outcoming geodesic segment span an angle θ, which has to be positive, at x.
Then we may find two points P1 , P2 on l near cut-locus of x such that the distance
of their preimages on the tangent space with induced metric by the exponential
map at x is bounded from below by θ, but clearly not on M . This is clearly a
contradiction for sufficiently small and the fact that the metric is quasi-isometric
to the Euclidean one.
(b). To consider the derivatives of the Bergman metric, again we consider the uniP
formizing squeezing coordinates and let Kx (z, w) = i fi (z)fi (w) be the coefficient
of the Bergman kernel on ϕx (M ) which is holomorphic in z and conjugate holomorphic in w. Let M be the set M equipped with the conjugate complex structure.
Writing wi = ui , we conclude that Kx (z, ū) is holomorphic on M × M with respect
to the complex structures on M and M respectively. The restriction K(z, w) to
w = z is precisely the potential for the Bergman metric.
Let A = B a2 × B a2 . Let D be a differential operator involving compositions
of the coordinate derivatives. By Generalized Cauchy Inequality, it follows easily
∂
that all the higher derivatives |[D ∂z∂ i ∂w
log K](zo , uo )| of the metric at the origin
j
are controlled up to a constant depending on D by |K(z, u)| for (z, u) lying on
the boundary ∂(A). Clearly |Kx (z, u)|2 6 Kx (z, z)Kx (u, u) by the Cauchy-Schwarz
Inequality for (z, u) ∈ ∂A.
In terms of the peak function fz at z ∈ ϕ(M ) mentioned before, we obtain
Z
1
2
|fz |2
Kx (z, z) = |fz | 6
vol(B a2 (z)) vol(B a (z))
2
1
6
,
vol(B a2 )
14
SAI-KEE YEUNG
since the L2 -norm of fz is 1. The same bound is applicable to Kx (u, u). Restricting
to the twisted diagonal given by u = z̄, it follows immediately that the curvature
tensor and all their derivatives are bounded with respect to the Euclidean metric
on B a2 (z) ⊂ M . As gB is uniformly quasi-isometric to go on vol(B a2 ), we conclude
that all the derivatives of the curvature are uniformly bounded for the Bergman
metric. As in part (a), the finishes the proof of (b).
Proposition 5. (a). M is Kähler-hyperbolic with respect to gKE .
(b). The same is true for gB .
The proof of the proposition depends on the following lemma.
Lemma 4. Let g = gKE . Fix x, y ∈ M . Let W be a (1, 0)-vector at 0 ∈ ϕy (M )
Let ϕy,x : ϕy (M ) → ϕx (M ) be the biholomorphic mapping given by ϕx ◦ ϕ−1
y . Then
|∂W log |J(ϕy,x )|2 |
√
g(W,W )
(0) 6 C for some constant C independent of x and y.
Proof Let z i , wj , i, j = 1, . . . , n be the local coordinates on ϕy (M ) and ϕx (M )
respectively. For simplicity, we also denote ϕy,x by ϕ since x and y are fixed in this
∂
∂
proof. Let us choose the coordinate at ϕx (M ) such that g( ∂w
i , ∂w j ) is diagonal at
ϕ(0).
It suffices for us to show that for each k = 1, . . . , n,
|
2
∂
k log |J(ϕy,x )| |
∂zq
g( ∂k , ∂k )
∂z
∂ z̄
(0) 6 C.
Clearly,
|J(ϕ)(z)|2 =
det(g( ∂z∂ i , ∂z∂ j ))(z)
∂
∂
det(g( ∂w
i , ∂w j ))(ϕ(z))
.
Hence
| ∂z∂k log |J(ϕy,x )|2 |
1
∂
∂
∂
q
(0) 6 [ q
| log | det(g( i , j ))||]z=0
k
∂z
∂z
∂z
∂
∂
∂
∂
g( ∂zk , ∂ z̄k )
g( ∂zk , ∂ z̄k )
+[ q
1
g( ∂z∂k , ∂∂z̄k
∂
∂
∂
| log | det(g( i ,
))(ϕ(z))||]z=0 .
k
j
∂z
∂w
∂w
)
For the first term, applying Schauder type estimates to the chart ϕy (M ) and
using the fact that g = gKE is Kähler-Einstein, we conclude as in the proof of
Proposition 3 that
1
∂
∂
∂
[| q
| log | det(g( i , j ))(z)||]z=0 < c1 ,
k
∂z
∂z
∂z
∂
∂
g( ∂zk , ∂ z̄k )
Note that quasi-isometry of g = gKE and the Euclidean metric is used here.
For the second term, we rewrite w = ϕ(z) and use Chain rule to rewrite
GEOMETRY OF DOMAINS WITH THE UNIFORM SQUEEZING PROPERTY
[q
=
1
g( ∂z∂k , ∂∂z̄k )
[ qP
∂
∂
∂
| log | det(g( i ,
))(ϕ(z))||]z=0
∂z k
∂w ∂wj
1
|
i
i,j
∂
∂
∂w
g( ∂w
i , ∂w j ) ∂z k
∂wj
∂z k
1
=
q
P
i
15
X ∂wl ∂
∂
∂
| log | det(g( i ,
))(ϕ(z))|]z=0
∂z k ∂wl
∂w ∂wj
l
[|
∂wi
∂
∂
2
g( ∂w
i , ∂w i )| ∂z k | (0)
X ∂wl
l
∂z k
(0)
∂
∂
∂
| log | det(g( i ,
))(w)|]w=ϕ(0) .
∂wl
∂w ∂wj
Applying now Schauder’s estimate to ϕx (M ) and using the fact that g is KählerEinstein and g is quasi-isometric to the Euclidean metric on ϕx (M ) again, we
∂
∂
∂
conclude that | ∂w
l | log | det(g( ∂w i , ∂w j ))(w)|w=ϕ(0) 6 c2 . Hence
1
q
P
i
|
i
∂
∂
∂w 2
g( ∂w
i , ∂w i )(ϕ(0))| ∂z k | (0)
∂wl
∂
∂
∂
(0) l | log | det(g( i ,
))(ϕ(0))| 6 c3
k
∂z
∂w
∂w ∂wj
and we conclude that
| ∂z∂k log |J(ϕy,x )|2 |
q
(0) 6 c1 + c3 .
g( ∂z∂k , ∂∂z̄k )
As k is arbitrary, this concludes the proof of the Lemma.
Remark Lemma 4 gives a proof of the part of Proposition 5 of [Y4] about boundedness of the first derivative of the Jacobian, showing that it corresponds to general
properties of uniform squeezing domains. The last sentence in Proposition 5 of [Y4]
that |J(Φy,x )| = 1 is incorrect and was not used in the subsequent arguments in
[Y4].
Proof of Proposition 5 (a) Fix a point x ∈ M and consider the uniform squeezing
coordinates ϕx associated to it, so that Ba (0) ⊂ ϕx (M ) ⊂ Bb (0). The Kähler form
ωKE associated to the Kähler-Einstein metric gKE,x satisfies
√
−1∂∂ log det gKE = cωKE
for some negative constant c. Left hand side of the above expression is independent of the particular coordinate ϕx that we are using. Note that the determinant
det(gKE ( ∂z∂ i , ∂z∂ j )) depends on the local coordinates ϕx (M ). We denote the quan√
tity by det(gKE,x ). In this way we may regard hx = −1∂ log det gKE,x as the
potential one form to satisfy dhx = ω, here note that hx depends on our fixed base
point x. Again, on B a2 (0), the equation is
∆gKE,x log det gKE,x = c.
Since gKE,x ∼ gK ∼ go on B a2 (0), the above equation is a strongly elliptic equation
with uniformly bounded coefficients. It follows from Proposition 3 that we have
a√bound |d log det gKE,x | 6 C for some uniform constant C. We conclude that
| −1∂ log det gKE,x | < C and hence
√
|hx |gKE,x = | −1∂ log det gKE,x (y)|gKE,x < C1
for some uniform constant C1 and all y ∈ B a2 (0).
16
SAI-KEE YEUNG
Now we need to worry about points y ∈ ϕx (M ) − B a2 (0). For such cases, we
consider the uniform squeezing coordinate ϕy as well, here we identify y with ϕ−1
x (y)
to simplify our notations. ϕy,x = ϕy ◦ ϕ−1
is
the
biholomorphism
from
ϕ
(M
) to
x
x
ϕy (M ). We have correspondingly det(gKE,x (z)) = det(gKE,y (w))|J(ϕy,x )|2 . Let
Y ∈ T0 (ϕy (M )) and X = (ϕy,x )∗ Y ∈ Tϕy,x (0) (ϕx (M ). Then
√
√
√
hx (X) =
−1∂X log det(gKE,x ) = −1∂Y log det(gKE,y ) + −1∂X log |J(ϕy,x )|2
√
= hy + −1∂X log |J(ϕy,x )|2 .
Clearly it follows from the earlier argument that |hy |gKE,y < c2 for some uniform
constant c2 . Lemma 4 also shows that |∂X log |J(ϕy,x )||2gKE,y < c3 . Proposition 3
for gKE now follows from combining these two estimates.
(b) Fix our point x ∈ M√ as before. The Kähler
P form of gB can be written as
ωB = dηx , where ηx = − −1Kx−1 ∂Kx and Kx j |fj (z)|2 by taking over unitary
basis of holomorphic functions on the Hilbert space of L2 -holomorphic functions on
ϕx (M ). Proposition 3b above implies that |ηx (y)|gB < c4 for some constant c4 and
every point y ∈ B a2 (0) ⊂ ϕy (M ).
For a point y ∈ ϕx (M ) − B a2 (0), we note that the potentials of gB satisfies
Kx = Ky |J(ϕy,x )|2 . Hence
√
ηx (X) = ηy (Y ) + −1∂X log |J(ϕy,x )|2
similar to the derivation in (a). From the previous paragraph, |ηy |gB < c5 for
some uniform constant c5 > 0. From Proposition 3b again, |∂X log |J(ϕy,x )(y)|2gB 6
c6 |∂X log |J(ϕy,x )(y)|2go | 6 c7 for some constant c7 > 0. (b) follows by combining
the previous two estimates.
Lemma 5. Let x be a fixed point on M . Expressed in terms of the uniform squeezing coordinate system ϕx (M ), | det(gKE )|−α for α sufficiently small is a bounded
plurisubharmonic exhaustion function on M.
Proof Denote by |gKE | = det(gKE ) the determinant of gKE in local coordinates.
Direct computation yields
√
−1∂∂(−|gKE |−α )
√
√
= −α(α + 1)|gKE |−α−2 −1∂|gKE | ∧ ∂|gKE | + α|gKE |−α−1 −1∂∂|gKE |
√
= α|gKE |−α −1[(α + 1)∂∂(log |gKE |) − ∂ log |gKE | ∧ ∂ log |gKE |].
Applying Proposition 5 and noting that log |gKE | is √
up to a constant the potential of
g√
KE on any realization of M as a bounded domain, | −1∂ log |gKE |∧∂ log |gKE )|| 6
c −1∂∂ log(|gKE |) for some constant c > 0. It suffices for us to choose α > 1c − 1
to conclude the proof of the lemma.
Proof of Theorem 2 (a) follows from Proposition 3. (b), (c) and (d) follows from
Proposition 4. (e) follows from Proposition 5. (f) follows from Lemma 5.
Remark We remark that finiteness in volume of gKE is in fact equivalent to the
quasiprojectiveness of our M . One direction is proved in the above corollary. For
the other direction , as assume that M1 = M/Γ is a quasi-projective manifold. We
may assume that M1 = M 1 − D for some normal crossing divisor D after resolution
of singularities if necessary. Hence neighborhoods of D in M1 are covered by union
GEOMETRY OF DOMAINS WITH THE UNIFORM SQUEEZING PROPERTY
17
of open sets Ui of the form ∆a1 × (∆∗1 )b , where ∆1 is a Poincaré disk of radius 1 and
∆∗1 is a punctured disk of radius 1.
Equip each such Ui = ∆a1 ×(∆∗1 )b , we consider a smaller open set Ur = ∆ar ×(∆∗r )b
for 0 < r < 1 with the restriction of the Poincaré metric gP on Ui and apply the
Schwarz Lemma of Mok-Yau [MY] to the embeddings of the inclusion of (Ui , gP )
into (M1 , gKE ), we conclude easily that the volume of (M1 , gKE ) is finite since the
volume of each (Ui , gP ) is finite.
§4 Examples
We are going to show that examples in Proposition 1 do satisfy the uniform
squeezing property.
Proof of Proposition 1 We will prove (a), (b), (d) and (e) first and leave the
proof of (c) to the end.
(a) Bounded homogeneous domain M in Cn .
Choose any point on M and translate the origin of Cn to that point. As it is
bounded, it is contained in a ball Bb (x). Since 0 lies in the interior of M , there
exists a ball of positive radius a such that Ba (0) ⊂ M. Hence 0 ∈ Ba (0) ⊂ M ⊂
Bb (x) ⊂ Cn . Let x be an arbitrary point on M . As M is homogeneous, there exists
a biholomorphism of M moving x to 0, here we realize M as a fixed domain in Cn .
Hence balls of the same radii provide uniform squeezing coordinates for all x ∈ M.
(c) Bounded domains which cover a compact Kähler manifold
Assume that M ⊂ Bc (xo ) ⊂ Cn is a bounded domain covering a compact Kähler
manifold N , where xo is a fixed point on M . Let A be a fundamental domain
of N in M . For each point x ∈ A, there exists a ball of radius rx such that
Brx (x) ⊂ M. Since A is relatively compact, r = inf x∈A rx > 0. It is then clear
that Ba (x) ⊂ M ⊂ Bb (x) for each x ∈ A. Since each point y ∈ M can be mapped
biholomorphically to some point in A by the deck transformation group, it is clear
that we get a (a, b) uniform squeezing coordinate.
(d) Teichmüller spaces Tg,n of compact Riemann surfaces of genus g with n punctures
This is a consequence of Bers Embedding Theorem described as follows (cf.
[Ga]). Let S be a Riemann surface of genus g with n punctures representing a
point x ∈ Tg,n . Denote by TS the Teichmüller space based at x. There exists an
N
N
is identified with the
embedding Φ : TS → CN , so that B N
1 ⊂ TS ⊂ B 3 , where C
2
2
space of holomorphic quadratic differentials based at S equipped with L∞ norm,
and Φ(x) = 0, where N = 3g − 3 + n.
Hence the charts associated to Bers embedding provide us the uniform squeezing
coordinates.
(b) Bounded smooth strongly convex domains
We give a step by step construction of the uniform squeezing coordinate systems.
(i) We observe the following fact. Suppose C1 = ∂Ba1 (x) and C2 = ∂Bb1 (y) are
two circles in C of radii a and b meeting tangentially at one point. Assume that
Ba1 (x) ⊂ Bb1 (y). Let w ∈ Ba1 (x) lying on the real line joining x and y. Then there
exists a Möbius f mapping C1 to itself, so that f is holomorphic on Bb1 (y), f (w) = 0
1
and f (C2 ) ⊂ B2b
(0).
18
SAI-KEE YEUNG
To see this, we may assume that x = 0 and a = 0 by rescaling. By a linear change
of coordinates, we may also assume that y = −b + 1 lies on the real axis of C. The
fact follows by inspecting the explicit Möbius transformation z → (z − w)/(1 − zw).
(ii) We claim the following fact. Suppose Ba (x) ⊂ Bb (y) are two balls in Cn
and ∂Ba (x) is tangential to ∂Bb (y) at a point q. Let w ∈ Ba (x) lying on the
real line joining x and y. Then there exists a Möbius transformation ψ of Ba (x),
so that ψ is biholomorphic on Ba (x), ψ is holomorphic on Bb (y), ψ(w) = 0 and
ψ(Bb (y)) ⊂ B2b (0).
To see this, after a linear change of coordinates, we may assume that the real
line joining x and y is defined by z2 = · · · zn = 0 and Im(z1 ) = 0. As in (i), we
may assume that x = 0 by an affine change change of coordinate, and a = 1 after
rescaling. Consider now the Möbius transformation given by
p
p
1 − |w|2
1 − |w|2
z1 − w
,
z2 , · · · ,
zn ).
ψ(z1 , · · · , zn ) = (
1 − z1 w 1 − z1 w
1 − z1 w
The same computation as in (i) establishes the claim.
(iii) We now proceed to construct the uniform squeezing coordinate system. We
are considering a C 2 -strongly convex domain M in Cn . Let p ∈ ∂M. Let Up be
a neighborhood of p. For a point q ∈ Up0 := ∂Mp ∩ Up , let NUp0 (q) be the real
line which is normal to ∂M at q with respect to the Euclidean metric. As ∂M
is C 2 -smooth and M is convex, there exist point xp,q , yp,q ∈ NUp0 (q), and positive
numbers ap,q and bp,q such that both ∂Bap,q (xp,q ) and ∂Bbp,q (yp,q ) are tangential
to ∂Mp0 at q and Bap,q (xp,q ) ⊂ M ⊂ Bbp,q (yp,q ).
Replacing Up by a slightly smaller relatively compact subset of itself if necessary,
we may assume that ap = lim inf q∈Up0 ap,q > 0 and bp = 2 lim supq∈Up0 bp,q < ∞. Let
x0p,q be the unique point on the normal line NUp0 (q) ∩ M at a distance ap from q.
Let Vp = ∪q∈Up0 Bap (x0p,q ). From the above construction and from the claim in (ii),
an (ap , bp )-uniform squeezing coordinate charts exists for Vp .
The union ∪p∈∂M Vp covers a neighborhood of ∂M. From compactness of ∂M, we
can choose a finite number of points p1 , · · · , pN on ∂M such that ∩N
i=1 Vpi covers a
neighborhood of ∂M. Let Vo be a relatively compact open subset of M containing
M − ∩N
i=1 Vpi so that {Vi }i=0,...,N gives a holomorphic covering of M. It is clear that
there exists 0 < a0 < b0 such that for each point z ∈ V0 , there exists a holomorphic
coordinate charts with Ba0 (z) ⊂ M ⊂ Bb0 (z). Let a = min(a0 , ai , 1 6 i 6 N ) and
b = max(b0 , bi , 1 6 i 6 N ). It follows from our construction that the balls of radii
a and b involved form an (a, b)-uniform squeezing coordinate system for M. Hence
strictly convex domain with C 2 boundary satisfies the uniform squeezing property.
This concludes the proof of Proposition 1.
§5 Geometric consequences
In this section, we give a proof of Corollary 1, Corollary 2 and Theorem 3 as
applications of Theorem 1 and 2. A proof for Theorem 4 is also explained.
Proof of Corollary 1 (a) and (c) follows from the argument in [Gr]. (b) is already
proved in [M], once we know that the manifold involved is Kähler-hyperbolic.
GEOMETRY OF DOMAINS WITH THE UNIFORM SQUEEZING PROPERTY
19
Proof of Corollary 2 (a). It is given that N = M/Γ is a compact complex
manifold. Since we have already proved that there exists a Kähler-Einstein of
negative scaler curvature on M which descends to N , it follows immediately that
the canonical line bundle is ample and hence the variety is of general type.
(b). From assumption, M/Γ has finite volume with respect to the Kähler-Einstein
metric gKE . From Theorem 2a, gKE is complete Kähler, with constant negative
Ricci curvature and bounded Riemannian sectional curvature. Hence we may apply
the results of [Y1], which relies on the earlier results of Mok-Zhong [MZ], to conclude
that M1 is quasi-projective.
We note that hi(2) (M1 , lKM1 ) = 0 for l ≥ 2 from L2 -estimates as in the corresponding proof of Kodaira’s Vanishing Theorem. Hence hi(2) (M1 , lKM1 ) = χ(M, lKM ).
It follows from a generalized form of the Riemann-Roch estimates of Demailly [De]
as proved in [NT] that
Z
ln
c1 (KM )n + o(lN ).
χ(M1 , lKM1 ) =
n! M
Here we note that c1 (KM ) is positive definite from construction. L2 -holomorphic
0
sections of lKM extends as a holomorphic section in HM
(M 1 , l(KM 1 + D)). This
1
2
follows from the fact that they extend as L -sections and hence cannot have poles
of order greater than 1 along any component of the compactifying divisor. We
0
(M 1 , l(KM 1 + D))) ≥ clN and hence that (M 1 , D) is of
conclude that dim(HM
1
log-general type. This concludes the proof of Corollary 2.
Proof of Theorem 3 It follows from Proposition 1 that a bounded domain which
is the universal covering of a complex manifold is equipped with a uniform squeezing coordinate. Hence from Theorem 1, it is pseudoconvex and hence a Stein
manifold. Furthermore, it supports a complete Kähler-Einstein gKE metric of negative scaler curvature. Since gKE is invariant under biholomorphism and hence
the deck transformations, it descends to N . Hence the canonical line bundle of
N is ample and the manifold is of general type. We denote by hp,q
(2),v (M ) the von
Neumann dimension of the space of L2 -harmonic (p, q)-forms on M with respect to
the Kähler-Einstein metric (cf. [At]). Corollary 1 implies that the von-Neumann
n,0
dimension hp,q
(2),v (M ) = 0 for p + q < n and h(2,v) (M ) > 0, which implies that the
corresponding Euler-Poincaré characteristics (−1)n χL2 ,v (M ) > 0. From Atiyah’s
Covering Index Theorem, χL2 ,v (M ) = χ(M/Γ). Hence (−1)n χ(M/Γ) > 0.
We may apply the same argument to the holomorphic line bundle 2K on M.
We use K to denote by the canonical line bundle on N and M. First of all
h0L2 ,v (M, 2K) > 0 by the usual L2 -estimates as used in [Y2], noting that gKE has
strictly negative Ricci curvature. The same L2 -estimates implies that hiL2 ,v (M, 2K) =
0 for i > 0. Atiyah’s Covering Index Theorem implies that χ(N, 2K) > 0. On the
other hand, from Kodaira’s Vanishing Theorem or L2 -estimates, we conclude that
hi (N, 2K) = 0 for i > 0. Hence h0 (N, 2K) = χ(N, 2K) > 0. This concludes the
proof of Theorem 3.
Proof of Corollary 3 From Theorem 1, we know that Ω is pseudoconvex. It was
proved by Mok-Yau in [MY] that a complete Kähler-Einstein metric of negative
1
sectional curvature exists and its volume form is bounded from below by d2 (− log
d)2
with respect to the Euclidean coordinates. Corollary 3 follows from the proof of
20
SAI-KEE YEUNG
Theorem 2 as the Bergman kernel is shown to be equivalent to the Kähler-Einstein
volume form. Note that both of them transforms under a coordinate change by the
same Jacobian determinant as in the proof of Lemma 2.
Proof of Corollary 4 Stehlé has proved in [St] the result that a locally trivial
holomorphic fiber space with hyperconvex fibers and Stein base is Stein. Corollary
3 follows immediately from Theorem 2f.
Remark There are many positive results to Serre’s problem, including the result
of Siu [Si] when the fibers have trivial first Betti number, the result of Mok [Mo]
when the fibers are Riemann surfaces, and the results of Diederich and Fornaess
[DF]. In general the problem has negative solution due to counterexamples such as
the one given by Skoda in [Sk].
Proof of Theorem 4 As explained in the last section, Bers Embedding gives rise to
a uniform squeezing coordinate system. All the results of Theorem 4a-e follow from
the earlier results of this paper under the sole assumption that a uniform squeezing
coordinate system exists for Tg,n , which is provided by Bers Embedding. Theorem
4f also follows from Corollary 2 if we accept that Mg,n is quasi-projective, which is
known classically by the well-known work of Baily for n = 0 and Knudsen-Mumford
for the case of n 6= 0 (cf. [KM]).
The exact formula for the Euler characteristic of M has already been obtained
by Harer-Zagier [HZ]. We just remark that using Kähler hyperbolicity and Atiyah’s
Covering Theorem as in Theorem 3, we may prove that Euler-Poincaré characteristic of M1 satisfies (−1)n χ(M1 ) > 0 as well. The only minor difference is that
M1 is now non-compact. However, it follows from Theorem 2 that (M1 , gKE ) has
finite geometry and hence the chopping argument of Cheeger-Gromov [CG] shows
that one may exhaust M1 by appropriate relatively compact sets so that the contribution from the boundary tends to 0 as one takes the limit on the exhaustion.
It follows that (−1)n χ(M1 ) > 0.
This concludes the proof of Theorem 4.
Remark It is known that gK , gC , gB , gKE , gT and gM are quasi-isometric on Mg,n ,
where gT is the Teichmüller metric and gM is a Kähler metric constructed in [Mc],
as is shown in [Y3]. It is also proved in [Mc] that any order of derivatives of gM is
bounded as well. Combining with Theorem 2c and the proof there, the difference
k∇gX11 ,···XN Rg1 − ∇gX21 ,···XN Rg2 kg1 is bounded for any g1 , g2 chosen among gB , gKE
and gM . Hence gB , gKE and gM are all comparable up to any order of derivatives.
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Mathematics Department, Purdue University, West Lafayette, IN 47907 USA
E-mail address: [email protected]